A pipe \(900 \mathrm{~m}\) long and \(200 \mathrm{~mm}\) diameter discharges water to atmosphere at a point \(10 \mathrm{~m}\) below the level of the inlet. With a pressure at inlet of \(40 \mathrm{kPa}\) above atmospheric the steady discharge from the end of the pipe is \(49 \mathrm{~L} \cdot \mathrm{s}^{-1}\). At a point half way along the pipe a tapping is then made from which water is to be drawn off at a rate of \(18 \mathrm{~L} \cdot \mathrm{s}^{-1}\). If conditions are such that the pipe is always full, to what value must the inlet pressure be raised so as to provide an unaltered discharge from the end of the pipe? (The friction factor may be assumed unaltered.)

Bernoulli's equation plays a vital role in fluid mechanics, governing the behavior of fluid flow. It describes the relationship between pressure, velocity, and elevation in a moving fluid. The equation is expressed as:
\[ P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant} \].
In this context, each term represents different forms of energy per unit volume in the fluid:

• \( P \) is the pressure energy.
• \( \frac{1}{2}\rho v^2 \) is the kinetic energy due to fluid velocity.
• \( \rho gh \) is the potential energy related to the elevation of the fluid.

When solving problems using Bernoulli's equation, you compare these energies at two different points along the flow path. For instance, in the given problem, we compare the energy at the pipe's inlet with the energy at the outlet to understand how pressure and elevation changes influence the flow. Knowing the inlet pressure, elevation difference ( \( h \)), and exit conditions (typically atmospheric pressure), allows us to solve for unknown variables like velocity.
Understanding how to apply Bernoulli's equation and recognizing each term's significance helps tremendously in fluid mechanics problem-solving.

The Darcy-Weisbach equation is crucial for analyzing head loss due to friction in pipes. In fluid mechanics, this equation helps estimate the energy lost as fluid flows through a pipeline. The equation is given by:
\[ h_f = f \frac{L}{D} \frac{v^2}{2 g} \],
where:

• \( h_f \) is the frictional head loss.
• \( f \) is the Darcy friction factor.
• \( L \) is the length of the pipe.
• \( D \) is the diameter of the pipe.
• \( v \) is the average velocity of the fluid.
• \( g \) is the acceleration due to gravity.

In the context of the problem, the frictional head loss is split into two segments of the pipe when a tapping point is added midway. The velocity of the fluid changes with the discharge rate introduced by the tapping point, thus altering the frictional losses. By applying the Darcy-Weisbach equation to each segment before and after the tapping point, we can determine the total head loss and ensure accurate pressure adjustments to maintain consistent discharge throughout the pipe length.
Understanding the Darcy-Weisbach equation and how to apply it is essential for calculating and optimizing fluid flow in pipelines, ensuring that the system operates efficiently.

The continuity equation is a fundamental principle in fluid mechanics, stating that the mass flow rate must be constant throughout a closed system. Mathematically, it is expressed as:
\[ Q = A v \] ,
where:

• \( Q \) is the volumetric flow rate.
• \( A \) is the cross-sectional area of the pipe.
• \( v \) is the flow velocity.

In simpler terms, for incompressible flow, the amount of fluid entering a section of the pipe must equal the amount of fluid leaving that section. If the pipe's diameter changes, the velocity must adjust to ensure the flow rate remains the same.

In the given problem, the continuity equation helps in understanding flow rates at different sections of the pipe. Initially, we have a steady discharge of 49 L/s. When water is drawn off at a midway point at a rate of 18 L/s, the continuity equation ensures that the remaining flow downstream accounts for the 49 L/s discharge at the pipe's end. Thus, we apply:
\[ Q_{in} = Q_{out} + Q_{tap} \rightarrow 0.049 \text{ m}^3/\text{s} = 0.031 \text{ m}^3/\text{s} + 0.018 \text{ m}^3/\text{s} \].
Ensuring the combined flow rates result in the original discharge at the outlet. Understanding and applying the continuity equation ensures fluid systems maintain consistent operations despite changes in the pipeline due to adjustments like tapping points or diameter variations.